In this paper we develop accelerated first-order methods for convex optimization with locally Lipschitz continuous gradient (LLCG), which is beyond the well-studied class of convex optimization with Lipschitz continuous gradient. In particular, we first consider unconstrained convex optimization with LLCG and propose accelerated proximal gradient (APG) methods for solving it. The proposed APG methods are equipped with a verifiable termination criterion and enjoy an operation complexity of ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ and ${\cal O}(\log \varepsilon^{-1})$ for finding an $\varepsilon$-residual solution of an unconstrained convex and strongly convex optimization problem, respectively. We then consider constrained convex optimization with LLCG and propose an first-order proximal augmented Lagrangian method for solving it by applying one of our proposed APG methods to approximately solve a sequence of proximal augmented Lagrangian subproblems. The resulting method is equipped with a verifiable termination criterion and enjoys an operation complexity of ${\cal O}(\varepsilon^{-1}\log \varepsilon^{-1})$ and ${\cal O}(\varepsilon^{-1/2}\log \varepsilon^{-1})$ for finding an $\varepsilon$-KKT solution of a constrained convex and strongly convex optimization problem, respectively. All the proposed methods in this paper are parameter-free or almost parameter-free except that the knowledge on convexity parameter is required. To the best of our knowledge, no prior studies were conducted to investigate accelerated first-order methods with complexity guarantees for convex optimization with LLCG. All the complexity results obtained in this paper are entirely new.
翻译:在本文中, 我们开发了与本地 Lipschitz 连续梯度( LLCG) 同步优化的加速一阶方法, 与本地的 Lipschitz 连续梯度( LLCG) 相比, 已经超越了经过仔细研究的类 convex 优化与 Lipschitz 连续梯度( LLCG) 。 特别是, 我们首先考虑与 LLCG 一起进行不受限制的 convex 优化, 并提出了解决这一问题的快速精度优化方法。 我们随后考虑与 LLCG 一起进行限制的 convex 优化, 并提议通过应用我们提议的 AGP 方法来解决它, 大约解决了 一种经我们研究后增加的纸质免费的一阶( log\\ vareplus) 子公司 第一次( Varrecialexlus comlus) 的亚值 。 因此, 在 Oral- 1 ralx 之前, 以可核查的 ral- ral ral or or or or or or or deal orx ormax drouplex 进行一个最高级的计算。